Knot contact homology as a planar limit of Chern-Simons theory

We prove a conjecture relating augmentation varieties to the large $N$ limit of Chern-Simons theory. Although this does not directly establish that the augmentation polynomial of a knot is the classical limit of a deformed $\hat{A}$-polynomial - as suggested by Aganagić and Vafa - it reduces the problem to characterizing certain algebraic properties of a module over the quantum torus, introduced in work of Gaiotto, Kannagi, and Sanjurjo. We term this the \emph{HOMFLYPT difference module}, which captures relations between the colored HOMFLYPT polynomials of different antisymmetric colorings. We demonstrate that the classical limit of this difference module for a knot is precisely the degree 0 abelianized knot contact homology of the knot, and we provide a natural extension of this result to links.